SIAM Review, Volume 66, Issue 2, Page 203-203, May 2024.
Inverse problems arise in various applications---for instance, in geoscience, biomedical science, or mining engineering, to mention just a few. The purpose is to recover an object or phenomenon from measured data which is typically subject to noise. The article “Computational Methods for Large-Scale Inverse Problems: A Survey on Hybrid Projection Methods,” by Julianne Chung and Silvia Gazzola, focuses on large, mainly linear, inverse problems. The mathematical modeling of such problems results in a linear system with a very large matrix $A \in \mathbb{R}^{m\times n}$ and a perturbed right-hand side. In some applications, it is not even possible to store the matrix, and thus algorithms which only use $A$ in the form of matrix-vector products $Ax$ or $A^Tx$ are the only choice. The article starts with two examples from image deblurring and tomographic reconstruction illustrating the challenges of inverse problems. It then presents the basic idea of regularization which consists of augmenting the model by additional information. Two variants of regularization methods are considered in detail, namely, variational and iterative methods. For variational methods it is crucial to know a good regularization parameter in advance. Unfortunately, its estimation can be expensive. On the other hand, iterative schemes, such as Krylov subspace methods, regularize by early termination of the iterations. Hybrid methods combine these two approaches leveraging the best features of each class. The paper focuses on hybrid projection methods. Here, one starts with a Krylov process in which the original problem is projected onto a low-dimensional subspace. The projected problem is then solved using a variational regularization method. The paper reviews the most relevant direct and iterative regularization techniques before it provides details on the two main building blocks of hybrid methods, namely, generating a subspace for the solution and solving the projected problem. It covers theoretical as well as numerical aspects of these schemes and also presents some extensions of hybrid methods: more general Tikhonov problems, nonstandard projection methods (enrichment, augmentation, recycling), $\ell_p$ regularization, Bayesian setting, and nonlinear problems. In addition, relevant software packages are provided. The presentation is very clear and the paper is also readable for those who are not experts in the field. Hence, it is valuable for everyone interested in large-scale inverse problems.

中文翻译：

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 调查与回顾


《SIAM 评论》，第 66 卷，第 2 期，第 203-203 页，2024 年 5 月。

逆问题出现在各种应用中——例如，在地球科学、生物医学或采矿工程中，仅举几例。目的是从通常受到噪声影响的测量数据中恢复对象或现象。 Julianne Chung 和 Silvia Gazzola 撰写的文章“大规模逆问题的计算方法：混合投影方法的调查”主要关注大型逆问题（主要是线性问题）。此类问题的数学建模会产生一个线性系统，该系统具有非常大的矩阵 $A \in \mathbb{R}^{m\times n}$ 和扰动的右侧。在某些应用中，甚至不可能存储矩阵，因此仅使用矩阵向量乘积$Ax$或$A^Tx$形式的$A$的算法是唯一的选择。本文从图像去模糊和断层扫描重建的两个示例开始，说明了逆问题的挑战。然后，它提出了正则化的基本思想，其中包括通过附加信息来增强模型。详细考虑了正则化方法的两种变体，即变分方法和迭代方法。对于变分方法，提前了解良好的正则化参数至关重要。不幸的是，它的估计可能很昂贵。另一方面，迭代方案（例如 Krylov 子空间方法）通过提前终止迭代来进行正则化。混合方法将这两种方法结合起来，利用每个类别的最佳功能。本文重点研究混合投影方法。在这里，我们从克雷洛夫过程开始，其中原始问题被投影到低维子空间上。然后使用变分正则化方法解决投影问题。 本文回顾了最相关的直接和迭代正则化技术，然后提供了混合方法的两个主要构建模块的详细信息，即生成解决方案的子空间和解决投影问题。它涵盖了这些方案的理论和数值方面，还介绍了混合方法的一些扩展：更一般的吉洪诺夫问题、非标准投影方法（丰富、增强、回收）、$\ell_p$正则化、贝叶斯设置和非线性问题。另外，还提供了相关的软件包。演示非常清晰，论文对于非该领域专家的人来说也可读。因此，它对于每个对大规模逆问题感兴趣的人来说都是有价值的。